When a signal is discrete in time, its Fourier transform becomes a continuous and periodic function in frequency. This relationship emerges from the duality between time and frequency domains. By interchanging the roles of time and frequency, the discrete-time signal can be interpreted as the Fourier series coefficients of a continuous, periodic frequency-domain function. In essence, the diagram below mirrors the structure of a Fourier series—only with time and frequency axes swapped.
More importantly, if the discrete time sequence is a sampled version of a continuous time signal that has no information above frequency Fn and the sampling rate Fs is greater than 2Fn (twice the highest frequency of the continuous signal), then the Fourier transform of the sampled signal will be a periodic version of the Fourier transform of the original, continuous waveform. All of the information in the continuous waveform can be derived from the sampled waveform.
The Fourier transform of a discrete, non-periodic signal is called the Discrete Time Fourier Transform or DTFT.
A discrete, non-periodic signal with a sample period of T has the following Fourier transform:
